Minimum and maximum uncertainty values in quantum harmonic oscillator

[cleggy]

Homework Statement

I have to find the minimum and maximum values of the uncertainty of $$\Delta$$x and specify the times after t=0 when these uncertainties apply.

Homework Equations

The wave function is Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))

and for all t is Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2)

The Attempt at a Solution

the expectation value <x> = 0 ( given in my text )

hence $$\Delta$$x= $$\sqrt{}<x^2>$$

using the sandwich integral

$$\int$$$$^{\infty}_{-\infty}$$ψ1$$\ast$$(x) x^2 ψ1(x) dx = $$\frac{3}{2}$$ a^2

$$\int$$$$^{\infty}_{-\infty}$$$$\psi$$3$$\ast$$(x) x^2 $$\psi$$3(x) dx = $$\frac{7}{2}$$a^2

$$\int$$$$^{\infty}_{-\infty}$$$$\psi$$3$$\ast$$(x) x^2 $$\psi$$1 (x) dx = $$\int$$$$^{\infty}_{-\infty}$$$$\psi$$$$\ast$$1(x) x^2 $$\psi$$3 dx = $$\sqrt{\frac{3}{2}}$$a^2

where a is the length parameter of the oscillator.


Where do I go from here?

 

[Cyosis]

You have calculated the probability density function correctly in a previous question you asked. Where has the time dependence disappeared? Secondly how does one find minimum/maximum values of a function generally?

 

[cleggy]

I don't know where the time dependence has gone

I'm not sure how to calculate the minimum and maximum values of a function?

 

[cleggy]

I've been going at this for so long now my mind is turning mushy

 

[Cyosis]

If you're following a course on quantum mechanics I'm pretty sure you know how to find the minimum/maximum of a function.

Hint: derivative

 

[cleggy]

so i have to do a first derivative test

 

[Cyosis]

First you need to find the correct time dependent expression for <x^2>. After that you take the derivative with respect to the appropriate variable and solve for said variable.

 

[cleggy]

I don't know how to find the time dependent expression for <x^2>

 

[Cyosis]

In your other thread on this topic you calculated the probability density function for this time dependent wave function. The one with a sine in it and a t etc? You must remember.
This is the correct "sandwich" expression to use. What you've calculated now is the time independent expectation value of x^2.

that should have been

|$$\Psi$$|^2 = [1/2][|$$\psi$$1|^2 + |$$\psi$$3|^2

+ 2$$\psi$$1$$\psi$$3sin(2wot)]